Problem: Factor the following expression: $-7$ $x^2+$ $22$ $x+$ $24$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(24)} &=& -168 \\ {a} + {b} &=& & & {22} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-168$ and add them together. Remember, since $-168$ is negative, one of the factors must be negative. The factors that add up to ${22}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${28}$ $ \begin{eqnarray} {ab} &=& ({-6})({28}) &=& -168 \\ {a} + {b} &=& {-6} + {28} &=& 22 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 {-6}x +{28}x +{24} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 {-6}x) + ({28}x +{24}) $ Factor out the common factors: $ x(-7x - 6) - 4(-7x - 6) $ Notice how $(-7x - 6)$ has become a common factor. Factor this out to find the answer. $(-7x - 6)(x - 4)$